Wilma and Greg were trying to solve the quadratic equation x^2 + bx + c = 0. Wilma wrote down the wrong value of b (but her value of c was correct), and found the roots to be 1 and 6. Greg wrote down the wrong value of c (but his value of b was correct), and found the roots to be -1 and -4. What are the actual roots of x^2 + bx + c = 0?

The roots are -2 and -3

Step-by-step explanation:

For a quadratic equation given as x² + bx + c, for roots x and y, the sum of roots is equal to the negation of the coefficient of the second term (i.e x + y = - b) while the products of the roots is equal to the coefficient of the second term (i.e x × y = c).

Since for the roots 1 and 6, only the value of c was correct, to get c we use the product of roots. Therefore, c = 1 × 6 = 6

Since for the roots -1 and -6, only the value of  b was correct, to get b we use the sum of roots. Therefore:

-b = - 1 + -4 = -5

b = 5

Since the quadratic equation is  x² + bx + c, substituting value of b and c and solving:

x² + 5x + 6 = 0

x² + 2x + 3x + 6 = 0

x(x + 2) + 3(x + 2) = 0

(x + 2)(x + 3) = 0

x = -2 or x = -3

The roots are -2 and -3

proportions are fractions that equal to each other. so if two fractions are the same, they are proportional.

let's check the answers to see if they are proportional.

question a is not proportional because you can't turn 1.5/3 to 3/9. if you try to multiply 1.5/3 by 2 or 3, you wont get 3/9. it's impossible to make the fractions the same.

question b is not proportional because you can't turn 22/33 to 4/6. 33 can't go into 6.

question c is proportional because when you multiply 2 and 3 by 71, you would get 142/213, and those fractions equal to each other

question d is proportional because when you divide 14 and 7 by 2, you would end up with 7/3.5, and those fractions equal to each other.